量子確率微分方程式の体系
–
物理と数学の狭間
–
斎藤健
(Takeshi SAITO),
有光敏彦(Toshihico ARIMITSU)
Institute of Physics, University of Tsukuba Ibaraki 305, Japan
1
Introduction
Studies of the Langevin equation for quantum systems werestarted by Senitzky [1], Lax
[2] and Haken [3]. They investigated the Langevin equation for a quantum mechanical
damped
harm.onic
oscillator. Inthe quantumLangevinequation, variables in both relevantand irrelevant systemsarestochastic operators. Putting the condition that theequal-time
canonicalcommutation relation should hold foralltimeevenfor stochastic operators, they
derived commutation relations
among
random force operators and their correlations.In their studies, Senitzky, Lax and Haken did not construct a representation space
explicitly. In quantum theory, observable operators do not have physical meaning until a
representation space isspecified. As was pointed out by Kubo [4], the quantum Langevin
equation is an operator equation defined on a total representation space, i.e., a space of
relevantsystem and of random forces. Anyrepresentationspaceof random force operators
had not been constructed by physicists.
Mathematicians such as Hudson, Parthasarathy and their $\mathrm{c}(\succ \mathrm{W}\mathrm{o}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{r}\mathrm{s}[5]- 1^{1}0]$
con-structed explicitly a representation space of random force operators. With the
represen-tation space, they realized a stochastic Schr\"odinger equation by analogy with the usual
quantum mechanics. A timeevolution generator satisfying the stochastic Schr\"odinger
equation was determined on the requirement of its unitarity, which is one of the
neces-sary conditions for construction of a canonical operator formalism. It seems that, for
mathematicians, aconstruction of the stochastic Liouville equation was out of their
con-siderations.
The stochastic Liouville equation was introduced first by Kubo $[11, 12]$ in order to
equa-tion is an equation ofmotion for a probability distribution function in phase space under
the influence of random forces. There had been a few attempts to extend the stochastic
Liouville equation to quantum systems. Gardiner et al. $[13, 14]$ and Dekker [15] derived
a quantum stochastic Liouville equation by obtaining, within the trace formalism, an
adjoint operator ofa time-evolution generator for quantum Langevin equation.
Further-more, Gardineret al. [16] rederived their stochastic Liouvilleequation on the basis of the
stochastic Schr\"odinger equation introduced by Hudson et al. by making use of the fact
that a density operator is a functional of wave functions. Within the density operator
formalism, it is impossible to extract an explicit form ofthe time.evolution generator
sat-isfying the stochastic Liouville equation, since the Liouville equation has entanglements
between relevant operators and a density operator due to commutators and
anticommu-tators amongthem. These difficulties prevent one from constructingacanonical operator
formalism based onthe stochastic Liouville equation.
Ontheother hand, within theframework of Non-EquilibriumThermoField Dynamics
(NETFD) $[17]-[19]$, aunified canonical operatorformalism of quantumstochastic
differen-tial equations was $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{I}^{\cdot}\mathrm{u}\mathrm{C}\mathrm{t}\mathrm{e}\mathrm{d}[20]-[30]$ on the basis of the stochastic Liouville equation.
The quantum stochastic differential equations include the quantum Langevin equation
and the quantum stochastic Liouville equation together with the corresponding quantum
master equation. Within NETFD, introducing two kinds of operators, with tilde and
withouttilde, the entanglements between relevant operators and adensity operator in the
stochastic Liouville equation can be disentangled. Therefore, one can extract the explicit
form of the time-evolution generator satisfying the stochastic Liouville equation, which
enables us to construct a unified canonical operator formalism.
In this paper, we will construct quantum Wiener processes by means of boson
annihi-lation and creation operators with their representation space extending mathematicians’
procedure and implanting it into NETFD. The thermal degree of freedom in the
quan-tum Wiener processes will be introduced by a Bogoliubov
transformation
in the thermalspace which is a representation space within NETFD. Then, starting from a stochastic
Shr\"odinger equation, we will show how one can obtain the time-evolution generator
sat-isfying a stochastic Liouville equation with the help of the fact that a density operator
is a functional of wave functions together with the principle of correspondence between
quantities in the thermal space and
in..
the Hilbert space. We will also show how oneequations on the basis of the time-evolutiongenerator.
2
Quantum
Wiener Processes
We will construct quantum Wiener processes at zero temperature according to Hudson
and Parthasarathy [5, 9, 10].
2.1
Fock Space
We introduce boson operators $b(t)$ and $b^{\mathrm{t}}(t)$ with $t\in[0, \infty)$ satisfying the canonical
commutation relations
$[b(t), b^{\uparrow}(t)]=\delta(t-s)$, $[b(t), b(s)]=0$, (1)
and define the vacuums $|0\rangle\rangle$ and $\langle\langle$$0|$ by
$b(t)|\mathrm{o}\rangle\rangle=0$, $\langle\langle$$0|b^{\uparrow}(t)=0$. (2)
We introduce $\mathrm{k}\mathrm{e}\mathrm{t}-$ and $\mathrm{b}\mathrm{r}\mathrm{a}$-vectors defined by
$|t_{1},$
$\cdots,$$t_{n} \rangle\rangle=\frac{1}{\sqrt{n!}}b^{\uparrow}(t1)\cdots b^{\mathrm{t}}(i)n|\mathrm{o}\rangle\rangle$, $\langle\langle$$t_{1},$$\cdots t_{n}|=\langle\langle 0|\frac{1}{\sqrt{n!}}b(t_{1})\cdots b(t_{n})$, (3)
which satisfy the orthonormalization condition
$\langle\langle t_{1}, \cdots, t_{n}|S1, \cdots, S\rangle m\rangle=\delta nm^{\frac{1}{n!}\sum-s}P\delta(i1-S_{1})\cdots\delta(t_{nn})$, (4)
and the completeness relation
$\sum_{n=0}^{\infty}(\prod_{l}^{n}\int_{0}^{\infty}dt\iota)|t_{1},$
$\cdots,$$t_{n}\rangle\rangle\langle\langle t_{1},$
$\cdots,$$t_{n}|=I.$ (5)
Here,$\Sigma_{P}$ indicates the summationoverallpossible permutations of$t_{1},$$\cdots$ ,$t_{n}$ with$s_{1},$$\cdots.s_{n}$
fixed. Therefore, the set of$\mathrm{k}\mathrm{e}\mathrm{t}$-vectors
$\{|t_{1}, \cdots , t_{n}\rangle\rangle\}$andthat of$\mathrm{b}\mathrm{r}\mathrm{a}$-vectors $\{\langle\langle t_{1,n}\ldots, t|\}$
form complete orthonormal systems. The vectorspace$\Gamma^{0}$ builton the complete
orthonor-mal basic vectors $|t_{1},$
$\cdots,$$t_{n}\rangle\rangle$ and $\langle\langle$$t_{1},$
$\cdots,$$t_{n}|$ is called the Fockspace*.
*Since annihilation and creation operators$b(t),$ $b^{\uparrow}(t)\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{s}\mathfrak{h}$’ bosonic canonical commutation relations (1), the vector space$\Gamma^{0}$ is also called the boson Fock spaceor thesymmetric Fock space [10].
2.2
Quantum
Wiener Processes
Let us define the operators $B_{t}$ and $B_{t}^{\mathrm{t}}$ on the
Fock
space $\Gamma^{0}$ by$B_{t}= \int_{0}^{t}dsb(s)$, $B_{\mathrm{t}}^{1}= \int_{0}^{\mathrm{t}}dsb^{\dagger}(S)$. (6)
Taking average of $B_{t},$ $B_{t}^{\mathrm{t}}$ and the product $B_{t}^{1}B_{s},$ $B_{t}B_{s}^{1}$ with respect to the vacuums
$|0\rangle\rangle$ and $\langle\langle$$0|$, we find that
$\langle\langle 0|B_{t}|\mathrm{o}\rangle\rangle=\langle\langle 0|B_{t}^{\dagger}|0\rangle\rangle=0$, (7)
$\langle\langle 0|B_{ts}\dagger B|0\rangle\rangle=0$, $\langle\langle 0|B_{t}B^{\uparrow\min}s|\mathrm{o}\rangle\rangle=(t, s)$, (8)
wherewe used (2) and (1). Since the correlations (7), (8) indicates that the operators $B_{t}$
and $B_{t}^{\mathrm{t}}$ on the Fock space $\Gamma^{0}$ can be interpreted as the Wiener process for a quantum
system, we call the operators the quantum Wiener processes. The processes $B_{t}$ and $B_{t}^{\dagger}$
are also called annihilation and creation processes, respectively $[9, 10]$.
2.3
Product Rules
Let us introduce the exponential $vector|e(f)\rangle\rangle,$ $\langle\langle$$e(f)|\in\Gamma^{0}$ by
$|e(f) \rangle\rangle=\exp[\int_{0}^{\infty}dtf(t)b^{\uparrow}(t)]|0\rangle\rangle$, $\langle\langle$$e(f)|= \langle\langle 0|\exp[\int_{0}^{\infty}dtf^{*}(t)b(t)]$ , (9)
where$f$ is anelement ofthe set $L^{2}$ of squareintegrable functionssatisfying$\int_{0}^{\infty_{d}}t|f(t)|^{2}<$
$\infty$
.
Since the sets $\{|e(f)\rangle\rangle|f\in L^{2}\}$ and $\{\langle\langle e(f)||f\in L^{2}\}$ of all exponential vectors arelinearly independent and total in theFockspace $\Gamma^{0}[10]$, any operatoronthe Fockspaceis
characterized by the action on the exponential vectors [5]. The annihilation and creation
operators $b(t)$ and $b^{\mathrm{t}}(t)$ are characterized by the relations
$b(t)|e(f)\rangle\rangle=f(t)|e(f)\rangle\rangle$, $\langle\langle$$e(f)|b^{\uparrow}(t)=\langle\langle e(f)|f*(t)$, (10)
respectively.
With the help of the properties (10), the quantumWienerprocesses $B_{\mathrm{t}}$ and $B_{t}^{\uparrow}$ defined
by (6) are characterized by the following relations:
$\langle\langle e(f)|dBt|e(f’)\rangle\rangle=f’(t)dt\langle\langle e(f)|e(f/)\rangle\rangle$, (11)
The products of the increments $dB_{t}=B_{t+\ ^{-}}B_{\mathrm{t}},$ $dB_{t}^{\uparrow}=B_{t+dt}^{1}-B^{\dagger}t$ and $dt$ are
charac-terized by thefollowing $\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}^{\uparrow_{:}}$
$\langle\langle e(f)|dBedB_{t}|e(f/)\rangle\rangle=O(dt^{2})$, (13)
$\langle\langle e(f)|dBtdB_{t}||e(f’)\rangle\rangle=dt\langle\langle e(f)|e(f/)\rangle\rangle+O(dt^{2})$, etc.. (14)
Taking into account of the terms of $O(dt)$ in $L^{2}$-space and neglecting the terms of
$o(dt)$, we have, from the matrix elements (11), (12) and (13), (14), the following product
rules [5]:
(15)
3
Quantum Wiener
Processes at
Finite
Tempera-tures
We will construct quantum Wiener processes at finite temperatures considering boson
operators on the thermal space, which is the representation space within NETFD. This
is the reconstruction of quantum Wiener processes at finite temperatures introduced by
Hudson and Lindsay $[7, 8]$, within the framework of NETFD.
3.1
Representation Space
We introduce the tilde operators $(\tilde{b}(t),\tilde{b}\dagger(t))$ on the tilde conjugate space $\tilde{\Gamma}^{0}$
associated with $(b(t), b^{\uparrow}(t))$ on $\Gamma^{0}$. Here, the tilde
$conjugation\sim \mathrm{i}\mathrm{s}$ defined by
$(A_{1}A_{2})^{\sim}=\tilde{A}_{1}\tilde{A}_{2}$, (16)
$(_{C_{1}A_{1}}+c_{22}A)^{\sim}=c^{*}\tilde{A}_{1}+1\tilde{A}_{2}c_{2}*$, (17)
$(\tilde{A})^{\sim}=A$, (18)
$(A\dagger)^{\sim}=\tilde{A}^{\uparrow}$, (19) $\uparrow_{O(X})$ indicates that
$\lim_{xarrow 0}\frac{O(x)}{x}=\alpha\neq 0$,
while$o(x)$ indicatesthat
where$A_{1},$ $A_{2}$ and$A$ arearbitrary operatorson $\Gamma^{0}$ and $c_{1}$ and $c_{2}$ are $\mathrm{c}$-numbers. Note that
the tilde conjugate space $\tilde{\Gamma}^{0}$
is the Fock space built on the $\mathrm{b}\mathrm{a}s$ic vectors made by cyclic
operations of $\tilde{b}^{\uparrow}(t)$ on the
vacuum
$|\tilde{0}\rangle\rangle$ and $\tilde{b}(t)$ on thevacuum
$\langle\langle$$\tilde{0}|$, where the vacuums $|\tilde{0}\rangle\rangle$ and $\langle\langle$$\tilde{0}|$ are defined by$\tilde{b}(t)|\tilde{\mathrm{o}}\rangle\rangle=0$, $\langle\langle$$\tilde{0}|\tilde{b}^{\uparrow}(t)=0$. (20)
Note that $|\tilde{0}\rangle\rangle$ and $\langle\langle\tilde{0}|$ are the tilde conjugate of $|0\rangle\rangle$ and $\langle\langle 0|$.
We consider the tensor product space
$\Gamma=\Gamma^{0}\otimes\tilde{\Gamma}0$. (21)
The
vacuum
states $|0$) and ($0|$ of $\Gamma$ is defined by$|0)=|0\rangle\rangle\otimes|\tilde{0}\rangle\rangle$, ($0|=\langle\langle 0|\otimes\langle\langle\tilde{0}|,$ (22)
which areinvariant under thetildeconjugation, i.e., $\{|0)\}^{\sim}=|0),$ $\{(0|\}^{\sim}=(0|$. Note that
$\Gamma$ is the Fock space built on the basic vectors made by cyclic operations of $(b^{\mathrm{t}}(t),\tilde{b}^{\uparrow}(t))$
on the vacuum $|0$) and $(b(i),\tilde{b}(t))$ on the vacuum ($0|$
.
In the following, we will use thenotational conventions such as
$b(t)\otimes\tilde{I}\Rightarrow b(t)$, $b^{\uparrow}(t)\otimes\tilde{I}\Rightarrow b^{\uparrow}(t)$, (23)
$I\otimes\tilde{b}(t)\Rightarrow\tilde{b}(t)$, $I\otimes\tilde{b}^{\uparrow}(t)\Rightarrow\tilde{b}^{\uparrow}(t)$, (24)
where $I$ and $\tilde{I}$
stand for identity operators on $\Gamma^{0}$ and $\tilde{\Gamma}^{0}$
, respectively. The annihilation
and creation operators $b(t),$ $b^{\mathrm{t}}(t),\tilde{b}(t),\tilde{b}^{\uparrow}(t)$ on $\Gamma$ satisfy the canonical commutation relations
$[b(t), b^{\mathrm{t}}(s)]=[\tilde{b}(t),\tilde{b}^{\uparrow}(S)]=\delta(t-s)$, (25)
$[b(t), b(s)]=[\tilde{b}(t),\tilde{b}(s)]=[b(t),\tilde{b}(s)]=[b(t),\tilde{b}^{\dagger}(s)]=0$. (26)
Note that the vacuums $|0$) and ($0|$ satisfy
$b(t)|0)=\tilde{b}(t)|0)=0$, ($0|b^{\uparrow}(t)=(0|\tilde{b}^{\dagger}(t)=0$. (27)
The thermal degree of freedom can be introduced by Bogoliubov
transformation
in $\Gamma$.
First, we require that the expectation value of $b^{\mathrm{t}}(t)b(s)$ should be
with areal positive number$\overline{n}$, where $\langle\cdots\rangle$ indicates the expectation with respect to some
states $|\rangle$ and $\langle$$|$. We findthat in order to insure the equation (28), it is sufficient to impose
the following conditions on the states $|\rangle$ and $\langle$$|$:
$b(t)| \rangle--\frac{\overline{n}}{1+\overline{n}}\tilde{b}^{\uparrow}(t)|\rangle$, $\langle$$|b^{\uparrow}(t)=\langle|\tilde{b}(t)$
.
(29)In fact, using the conditions (29), we have
$\langle b^{\uparrow}(t)b(s)\rangle=\frac{\overline{n}}{1+\overline{n}}\{\langle|b\dagger(t)b(s)|\rangle+\delta(t-s)\}$, (30)
which leads to (28). We call the states $|\rangle$ and $\langle$$|$ the thermal $ket$-vacuumand the thermal
$bra$-vacuum, respectively, and the conditions (29) the thermal state conditions for the
thermal ket- and bra-vacuums.
We introduce annihilation operators $c(t),\tilde{c}(t)$ and creation operators $c^{*}(t),\tilde{c}^{*}(t)$ for
the thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum
$|\rangle$ satisfying
$c(t)|\rangle=\tilde{c}(t)|\rangle=0$, $\langle$$|c^{*}(t)=\langle|\tilde{c}^{*}(t)=0$, (31)
and the canonical commutation relations
$[c(t), c^{*}(s)]=[\tilde{c}(t),\tilde{C}^{*}(S)]=\delta(t-S)$, (32)
$[c(t), C(\mathit{8})]=[\tilde{c}(t),\tilde{C}(s)]=[C(t),\tilde{c}(s)]=[c(t),\tilde{c}^{*}(_{S})]=0$, (33)
$[c^{*}(t), c^{*}(s)]=[\tilde{c}^{*}(t),\tilde{c}^{*}(s)]=[c^{*}(t),\tilde{c}(s)]=[c^{*}(t),\tilde{c}^{*}(s)]=0$. (34)
Recalling the thermal state conditions (29), we see that such operators $c(t),$ $c^{*}(t)$ and
their tilde conjugates are related to $b(t),$ $b^{\mathrm{t}}(t)$ and their tilde conjugates through the
transformation
$=(\tilde{b}^{\uparrow}(t)b(t))$
.
(35)The transformation (35) is called the Bogoliubov $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}_{0}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\ddagger[31]$. The Bogoliubov
transformation is the canonical onesuchthat the canonical commutation relations do not
change under this transformation.
Let the boson Fock space built on the basic ket- and $\mathrm{b}\mathrm{r}\mathrm{a}$-vectors made by cyclic
operations of$(c^{*}(t),\tilde{C}^{*}(t))$ onthe thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum
$|\rangle$ and of $(c(t),\tilde{c}(t))$ on the thermal $\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum $\langle$$|$, be denoted by
$\Gamma^{\beta}$.
\ddagger In the expression of the $\mathrm{B}\mathrm{o}_{6}^{\sigma}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{u}\mathrm{b}\mathrm{o}$transformation, there is a freedom of normalization constant.
Note that the Bogoliubov transformation (35) is $\mathrm{r}\mathrm{e}\backslash \mathrm{v}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{n}$ as
$c(t)=U_{B}^{-1}b(t)U_{B}$, $\tilde{c}^{*}(t)=U_{B}^{-1}\tilde{b}^{\dagger}(t)UB$, (36)
where
$U_{B}= \exp[-\overline{n}\int_{0}^{\infty}dtb^{\dagger\uparrow}(t)\tilde{b}(t)]\exp[\int_{0}^{\infty}dtb(t)\tilde{b}(t)]$ , (37)
$U_{B}^{-1}= \exp[-\int_{0}^{\infty}dtb(t)\tilde{b}(t)]\exp[\overline{n}\int_{0}^{\infty}dtb\dagger(t)\tilde{b}\uparrow(t)]$ . (38)
The equations (36) togetherwith the properties (27) and (31) givethe relations between
the thermal vacuums in $\Gamma^{\beta}$ and the vacuums in $\Gamma$ as follows:
$|\rangle=U_{B}^{-1}|\mathrm{o})$, $\langle$$|=(0|U_{B}.$ (39)
Using the formula ofthe Liealgebra ofSU(I,I) group $[32]_{-[}35]$, we can rewrite $U_{B}^{-1}$ as
normal ordered product
$U_{B}^{-1}= \exp[\frac{\overline{n}}{1+\overline{n}}\int_{0}^{\infty}dtb\dagger(t)\tilde{b}|(t)]$
$\cross\exp[-\ln(1+\overline{n})\int_{0}^{\infty}dt\{b^{\uparrow}(t)b(t)+\tilde{b}^{\uparrow}(t)\tilde{b}(t)+\delta(0)\}]$
$\cross\exp[-\frac{1}{1+\overline{n}}\int_{0}^{\infty}dtb(t)\tilde{b}(t)]$
.
(40)Here, $\delta(0)$ is the delta function $\delta(t)$ with $t=0$. The equations (39) and (40) together
with the property (27) give
$| \rangle=\exp[-\delta(0)\ln(1+\overline{n})\int_{0}^{\infty}dt]\exp[\frac{\overline{n}}{1+\overline{n}}\int_{0}^{\infty}dtb^{\uparrow}(t)\tilde{b}^{\mathrm{t}}(t)]|0)$. (41)
Since $\delta(0)=\infty$, the equation (41) shows that in the thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum
$|\rangle$, infinite
numberof the $(b(t),\tilde{b}(t))$-pairsarecondensed and the Fockspace $\Gamma^{\beta}$
is inequivalent to the
Fock space $\Gamma$ in the sense that any vector in $\Gamma^{\beta}$ can not be written as a superposition of
vectors in $\Gamma$ and vice versa.
On
the other hand, the equation (39) together with the expression (37) of $U_{B}$ gives$\langle|=(\mathrm{O}|\exp[-\overline{n}\int_{0}^{\infty}dtb^{\uparrow(t})\tilde{b}\dagger(t)]\exp[\int_{0}^{\infty}dtb(t)\tilde{b}(t)]$
$=( \mathrm{O}|\exp[\int_{0}^{\infty}dtb(t)\tilde{b}(t)]$ , (42)
where we used the property (27). We see that the equation (42) is consistent wvith the
thermal state condition (29) of the $\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum. In fact, using the equation (42) and the
3.2
Quantum Wiener
Processes
at Finite Temperatures
Quantum Wiener processes at finite temperatures are defined by the operators
$B_{t}= \int_{0}^{t}dsb(s)$, $B_{t}^{\dagger}= \int_{0}^{t}d_{S}b\dagger(S)$, (43)
and their tilde.conjugates represented in the Fock space $\Gamma^{\beta}$. The explicit
representa-tions of the processes $B_{t},$ $B_{t}^{1},\tilde{B}_{t}$ and $\tilde{B}_{t}^{\uparrow}$ in $\Gamma^{\beta}$ are given in terms of the Bogoliubov
transformation (35) by
$B_{t}= \int_{0}^{t}dS[c(s)+\overline{n}\tilde{c}^{*}(s)]=C_{t}+\overline{n}\tilde{C}_{t}^{*}$, (44)
$B_{t}^{1}= \int_{0}^{t}ds[\tilde{c}(s)+(1+\overline{n})c^{*}(s)]=\tilde{C}_{t}+(1+\overline{n})C_{t}^{*}$, (45)
and their tilde conjugates, where $C_{t},$ $C_{\ell}^{*},\tilde{C}_{t}$ and $\tilde{C}_{t}^{*}$ are the annihilation and creation
processes in $\Gamma^{\beta}$ defined
by
$C_{t}= \int_{0}^{t}dsc(s)$, $C_{t}^{*}= \int_{0}^{t}dsC(*s)$, (46)
and their tilde-conjugates.
Any operator in the Fock space $\Gamma^{\beta}$ can be characterized
by the exponential vectors
$|e(f, g)\rangle,$ $\langle$$e(f,g)|$ in
$\Gamma^{\beta}$ with
$f,$ $g\in L^{2}$ defined by
$|e(f,g) \rangle=\exp[\int_{0}^{\infty}dt\{f(t)C^{*}(t)+g^{*}(s)_{\tilde{C}}\#(s)\}]|\rangle$, (47)
$\langle$$e(f,g)|= \langle|\exp[\int_{0}^{\infty}dt\{f*(t)_{C(}t)+g(S)\tilde{C}(s)\}]$ , (48)
which satisfy the following relations:
$c(t)|e(f,g)\rangle=f(t)|e(f,g)\rangle$, $\langle$$e(f,g)|c(\# t)=\langle e(f,g)|f*(t)$, (49)
$\tilde{c}(t)|e(f,g)\rangle=g^{*}(t)|e(f,g)\rangle$, $\langle$$e(f,g)|\tilde{c}( t)=\langle e(f,g)|g(t)$. (50)
As the case of the construction of annihilation and creation processes $B_{t},$ $B_{t}^{\uparrow}$ in Fock
space $\Gamma^{0}$, the evaluation of matrix elements of the products ofthe increments
$dC_{t},$ $dC_{t}^{*}$,
$d\tilde{C}_{\mathrm{t}},$ $d\tilde{C}_{t}^{\mathrm{a}_{\mathrm{a}}}\mathrm{n}\mathrm{d}dt$ between exponential vectors $|e(f,g)\rangle$ and $\langle$$e(f,g)|$ with the help of the
properties (49), (50) gives the productrules of the increments $dC_{t},$ $dc_{t’ t}^{\mathrm{i}}d\tilde{c}_{t},$$d\tilde{C}*_{\mathrm{a}}\mathrm{n}\mathrm{d}dt$,
which are summarized as the following table.
By
means
of the expressions (44), (45) and their tilde conjugates and the product rules(51), we can evaluate the products of the increments of $dB_{t},$ $dB_{\mathrm{t}}^{\mathrm{t}},$ $d\tilde{B}_{t},$ $d\tilde{B}_{t}^{1}$ and $dt$ and
obtain the product rules summarized as follows:
(52)
Using the equations (44), (45) and their tilde conjugates, the commutation relation
(32) and the properties (31) of the thermal vacuums, we obtain the correlations of the
increments $dB_{t},$ $dB_{t}^{\uparrow},$ $d\tilde{B}_{\mathrm{t}}$ and $d\tilde{B}_{t}^{1}$ with respect to the thermal vacuums $|\rangle$ and $\langle$$|$ as
follows:
$\langle dB_{\mathrm{t}}\rangle=\langle dB_{t}^{\uparrow\rangle}=\langle d\tilde{B}_{t}\rangle=\langle d\tilde{B}_{t}^{1}\rangle=0$, (53)
$\langle dB_{t\rangle}^{\dagger_{dB}}S=\langle d\tilde{B}_{t\rangle}^{\dagger_{d\tilde{B}}}s=\langle dB_{t}d\tilde{B}_{S}\rangle=\langle d\tilde{B}_{t}dB_{S\rangle}=\overline{n}\delta(t-s)dtds$ , (54) $\langle dB_{t}dB_{s}^{\mathrm{t}}\rangle=\langle d\tilde{B}_{t}d\tilde{B}_{s}^{1}\rangle=\langle dB_{t}^{1}d\tilde{B}_{s}^{1\rangle}=\langle d\tilde{B}_{t\rangle}^{||}dB_{s}=(1+\overline{n})\delta(t-s)dtds$, (55)
(others) $=0$, (56)
where $\langle\cdots\rangle\equiv\langle|\cdots|\rangle$. From the correlations (53)$-(56)$, we see that putting $\overline{n}$ to the
Planck distribution given by
$\overline{n}=\frac{1}{\mathrm{e}^{\beta\omega}-1}$, (57)
with some positive number $\omega$ and the inverse of the temperature, $\beta=1/T$, the quantum
Wiener processes $B_{t},$ $B_{t}^{\mathrm{t}}$
are
essentially equivalent to those introduced in the problem ofquantum optics [3].
4
Quantum Stochastic Calculus
On the basis of the quantum Wienerprocesses at finite temperatures, we will investigate
thequantum stochastic calculus.
4.1
Adapted
Processes
The Fock space $\Gamma^{\beta}$ is decomposed as
$\Gamma^{\beta\beta_{\otimes}\beta}=\Gamma\Gamma t](t$
’ (58)
in which for$f,g\in L^{2}$,
$|e(f,g)\rangle=|e(ft],gt1)\rangle\otimes|e(f(t,g(t)\rangle,$ $(e(f,g)|=\langle e(ft],g_{t]})|\otimes\langle e(f_{(g_{(}t}t,)|$, (59)
whereweset
$f_{t1^{=f}]}\chi_{\mathrm{t}}$, $f_{(t}=f\chi(\mathrm{t},$ (60)
and assumed that
$|\rangle=|t]\rangle\otimes|_{(t}\rangle$, $\langle$$|=\langle_{t]}|\otimes\langle_{(t}|$. (61)
Here, $\chi_{t}$] and $\chi_{(t}$
are
defined by$\chi_{t]}(s)=\theta(t-s)$, $\chi_{(t}(s)=\theta(s-t)$, for $t,$$s>0$
.
(62)Note that $\Gamma_{t]}^{\beta}$ is the boson Fock space built on the vacuums $|_{t]}\rangle$ and $\langle_{t]}|$, while $\Gamma_{(t]}^{\beta}$ is the
boson Fock space built on the vacuums $|_{(t}\rangle$ and $\langle_{(t}|$
.
The quantum Wiener processes $B_{t}$,$B_{t}^{\mathrm{t}},\tilde{B}_{t},\tilde{B}_{t}^{1}$ are operators on the space $\Gamma_{t]}^{\beta}$.
Let us consider a tensor product space $\mathcal{H}_{S}\otimes\Gamma^{\beta}$ where $\mathcal{H}_{S}$ is a certain vector space. For$\mathrm{t}’ \mathrm{n}\mathrm{e}$sakeof notational convenience, weidentify the quantumWienerprocesses $B_{t},$ $B_{t}^{1}$, $\tilde{B}_{t},\tilde{B}_{t}^{\uparrow}$ in
$\Gamma_{t]}^{\beta}$ with their ampliations to $\mathcal{H}_{S}\otimes\Gamma^{\beta}$, i.e.
$I_{S}\otimes B_{t}\otimes I_{(t}\Rightarrow B_{t}$, $I_{S}\otimes B_{t}^{\mathrm{t}}\otimes I_{(t}\Rightarrow B_{t}^{\mathrm{t}}$, (63)
$I_{S}\otimes\tilde{B}_{t}\otimes I_{(t}\Rightarrow\tilde{B}_{t}$, $I_{S}\otimes\tilde{B}_{t}^{\mathrm{t}}\otimes I_{(t}\Rightarrow\tilde{B}_{t}^{1}$, (64)
where $I_{S}$ and $I_{(t}$ are the identity operators on $\mathcal{H}_{S}$ and $\Gamma_{(t}^{\beta}$, respectively.
An adapted process $F_{\mathrm{t}}$ is defined by
$F_{t}=F_{t}0\otimes I(t,$ (65)
where $F_{t}^{0}$ is an operator on $\mathcal{H}_{S}\otimes\Gamma_{t]}^{\beta}$. The quantum Wienerprocesses $B_{t},$ $B_{t}\dagger,\tilde{B}_{l},\tilde{B}_{t}^{\uparrow}$ are
adapted. For the adapted process $F_{t}$, we have
$[F_{t}, dB_{t}]=[F_{t}, dB_{t}^{\mathrm{t}}]=[F_{t}, d\tilde{B}_{t}]=[F_{t}, d\tilde{B}_{t}^{\mathrm{t}}]=0$. (66)
4.2
Quantum
Stochastic
Calculus
Let $X_{t}$ denote an arbitrary adapted process in $\mathcal{H}_{S}\otimes\Gamma^{\beta}$ and $Q_{t}$ denote quantum Wiener
Quantum stochastic integrals ofthe Ito type are defined by
$\int_{0}^{T}X_{t}dQt\equiv\lim\sum_{=i0}^{I-1}xt:^{\Delta Qi}t=\lim\sum_{0i=}^{I-1}Xt.\cdot(Q_{t}\dot{.}+1^{-}Q_{t_{i}})$, (67)
and
$\int_{0}^{\tau_{dQ}}tx_{t}\equiv\lim\sum_{=i0}^{I-1}\Delta Q_{t.t}x$
.
$= \lim\sum_{i=0}^{I-1}(Qt.\cdot+1-Qt:)X_{t}:$’ (68)
while those of the Stratonovich type are defined by
$\int_{0}^{\tau_{X_{t}\circ dQ_{t}}}\equiv\lim\sum_{i=0}^{I}\frac{X_{t}.+1^{+}x_{t}}{2}-1,$$\Delta Q_{t_{i}}=\lim\sum^{-1}Ii=0^{\cdot}\frac{X_{t}.+1^{+}x_{t}}{2}.(Q_{t}.+1-Q_{t}:)$, (69)
and
$\int_{0}^{\tau_{dQ_{t}\equiv\lim}}\circ Xt\sum_{=i0}^{I}\Delta Qt-1:.\frac{X_{t}.+1^{+}x_{t}}{2}.\cdot=\lim\sum_{i=0}^{I}(Q_{t}:+1^{-}Q-1t:)\frac{X_{t_{i+1}}+X_{t}}{2}\dot{.}$ . (70)
Here, the notation $\lim$ indicates taking the limit
$\Delta tarrow+0$, $Iarrow+\infty$, (71)
keeping$T=I\Delta t$ fixed. Note thatquantum stochastic integrals both of the Ito and of the
Stratonovich types are adapted processes.
We introduce the differential notations of the stochastic integrals (67), (68) and (69),
(70) as $X_{t}dQ_{\mathrm{t}t}\equiv X(Q_{t}+dt-Q_{t})$, (72) $dQ_{t}X_{t}\equiv(Q_{tdl}+-Qt)Xt$, (73) and $X_{t} \circ dQt\equiv\frac{X_{t+dt}+x_{t}}{2}(Q_{t+dt}-Q_{t})$, (74) $dQ_{t^{\circ x_{t}}} \equiv(Qt+dt-Q_{\mathrm{t}})\frac{X_{t+dt}+x_{t}}{2}$. (75)
We call (72) and (73) the products of the Ito type, whereas we refer to (74) and (75) as the products ofthe Stratonovich type.
From the property (66), we have in the stochastic calculus ofthe Ito type
$[X_{t}, dQ_{t}]=0$, (76)
which leads to
Inaddition, usingtheproperty (61) ofthe thermal vacuums and theproperty ofquantum
$\mathrm{s}.\mathrm{t}.$
o..chastic
processes$\langle|.\dot{Q}_{t}|\rangle=0$, (78)
we have
$\langle|dQ_{t}x_{t}|\rangle=\langle|X_{t}dQt|\rangle--\langle_{t]}|x_{t}|t1\rangle\langle(t|dQ_{t}|(t\rangle=0,$ (79)
which indicates that there is no correlation between $X_{t}$ and $dQ_{t}$
.
It should bepointed out that while the increment $dQ_{t}$ commutes with $X_{t}$, it does not
commute with $X_{t+dt}$
.
Therefore, in the stochastic calculus of the Stratonovich type, thecommutation relation of$X_{t}$ and $dQ_{t}$ ofthe Stratonovich type defined by
$[X_{t}\circ, dQ_{t}]\equiv X_{t}\circ dQ_{t}-dQ_{t}\mathrm{o}X_{t}$, (80)
does not equal zero, i.e.
$[X_{t^{\mathrm{O}}}, dQ_{t}]\neq 0$, (81)
which leads to
$\int_{0}^{T}X_{t}\circ dQt\neq\int_{0}^{T}dQ_{t}\circ X_{t}$. (82)
Moreover, in contrast with thecase ofthe Ito type,
$\langle|X_{t}\circ dQ_{t}|\rangle\neq 0$, $\langle|dQ_{t}\circ Xt|\rangle\neq 0$. (83)
Substituting $X_{t+dt}=X_{t}+dX_{t}$ into (74) and (75), we obtain
$X_{t} \circ dQ_{t}=X_{t}dQ_{t}+\frac{1}{2}dX_{t}dQ_{t}$, (84)
and
$dQ_{t^{\circ X}t}=dQ_{t}X_{t}+ \frac{1}{2}dQ_{t1}dx$, (85)
which give the relations between the products of the Ito and the Stratonovich types.
4.3
Quantum
Ito’s Formula
We introduce an operator $N_{t}$ constructed by using the stochastic integrals oftheIto type
as
$N_{T}= \int_{0}^{T}(F_{t}dB_{t}+G_{t}dB_{t}^{\uparrow}+J_{t}d\tilde{B}_{t}+K_{t}d\tilde{B}_{t}^{\uparrow}+H_{t}dt)$, (86)
where $F_{t},$ $G_{t},$ $H_{t},$ $J_{t},$ $K_{t}$ are adapted processes. The operator $N_{t}$ is an adapted process
which satisfies
The differential notation of $N_{t}$ is given by
$dN_{t}=F_{\mathrm{t}}dB_{t}+G_{t}dB_{t}^{1}+J_{t}d\tilde{B}_{t}+K_{\mathrm{t}}d\tilde{B}_{\mathrm{t}}^{\mathrm{t}}+H_{t}dt$
.
(88)It should be noted that, for anarbitrary adapted process $X_{t}$, we have in general
$[X_{\mathrm{t}}, dN_{t}]\neq 0$, (89)
because $dN_{t}$ includes not only the increments $dB_{t},$ $dB_{t’ t}^{1}d\tilde{B},$ $d\tilde{B}_{t}^{\uparrow}$ but also adapted $\mathrm{p}\mathrm{r}(\succ$
cesses
$F_{t},$ $G_{t},$ $J_{t},$ $K_{t},$ $H_{t}$. Furthermore, for the adapted process $X_{t}$ and the increment$dN_{t}$, the property as (79) does not hold because of the term
$H_{\mathrm{t}}dt^{\S}$, i.e.
$\langle|X_{t}dN_{t}|\rangle=\langle|X_{t}H_{t}|\rangle dt\neq 0$, $\langle|dN_{\mathrm{t}}Xt|\rangle=\langle|H_{t}X_{t}|\rangle dt\neq 0$. (90)
For stochastic integrals, quantum $Ito’ s$
formula
holds [5, 9, 10]:Theorem 4.1 (Quantum Ito’s Formula) We set
$N_{T}= \int_{0}^{T}(F_{t}dB_{t}+G_{t}dB_{t}^{\dagger}+J_{t}d\tilde{B}_{\mathrm{t}}+K_{t}d\tilde{B}_{t}^{1}+H_{t}dt)$ , (91)
$N_{T}’= \int_{0}^{T}(F_{t}’dB_{t}+G_{t}’dB_{t}^{\uparrow}+J_{t}’d\tilde{B}_{\mathrm{t}}+K_{t}’d\tilde{B}_{t}^{\dagger}+H_{t}’dt)$ , (92)
where $F_{t},$ $G_{t},$ $H_{t},$ $J_{t},$ $K_{t},$ $F_{t}’,$ $G_{t}’,$ $H_{t}’,$ $J_{t}’,$ $K_{t}’$ are adapted $processe\mathit{8}$. The
differential
notations
of
$N_{t}$ and $N_{t}’$ are given by$d\Lambda_{t}^{T}=F_{t}dB_{\mathrm{t}}+G_{t}dB_{\iota}^{\dagger}+J_{t}d\tilde{B}_{t}+K_{t}d\tilde{B}_{\mathrm{t}}\dagger+H_{t}dt$, (93)
and
$dN_{t}’=F_{t}’dB_{t}+G_{t}’dB_{\mathrm{t}}^{1}+J_{t}’d\tilde{B}_{t}+K_{t}’d\tilde{B}_{t}^{\dagger}+H_{t}’dt$, (94)
respectively. Then, the
differential of
the product $\mathit{1}\backslash _{\mathrm{t}}N_{t}’$ can be evaluated by meansof
theformvla
$d(N_{t}N’t)=dN_{t}\cdot N_{t}’+N_{\mathrm{t}}\cdot d\underline,\mathrm{V}_{t}’+dN_{t}dN_{t}’$, (95)
with the property (66)
of
adapted processes and the product rules (52).Making useof the relations (84) and (85) between the Ito and the Stratonovich
prod-ucts, we have
$N_{\mathrm{t}} \circ dN_{t}^{;}=N_{t}dN_{t}’+\frac{1}{2}dN_{t}dN_{t}’$, (96)
\S When$H_{t}=0,$ $dN_{t}$ satisfies $\langle\langle 0|x_{tt}dN|\mathrm{o}\rangle\rangle=\langle\langle 0|d\mathit{1}\mathrm{V}tx\iota|0\rangle)=0$, although$dN_{t}$ still does not commute
and
$dN_{t} \circ Nt/=dN_{\mathrm{t}}N_{t}’+\frac{1}{2}dN_{t}dN_{t}’$
.
(97)Therefore, we findthatquantum Ito’sformula(95) is expressed in terms oftheStratonovich
products as
$d(N_{t}N_{t}’)=dN_{t}\circ N_{t}/+N_{t}\circ dN_{t}/$, (98)
which is identical to the well-known formula ofthe ordinary differential calculus.
5
Stochastic
Schr\"odinger
Equation
In this section, we consider the stochastic Schr\"odinger equation investigated by Hudson
and Lindsay [7].
5.1
The
Ito Type
We consider a boson system which is described by the operators $a,$
$a^{\uparrow}$ on a Hilbert
space
$\mathcal{H}_{S}^{0}$ satisfying the commutation relations
$[a, a^{\dagger}]=1$, $[a, a]=0$, (99)
and which interacts with a reservoir at finite temperatures. Let us suppose that the
effect of the reservoir on the system is taken into account by the random force operators
represented by the quantum Wiener processes at finite temperatures constructed on the
Fock space $\Gamma^{\beta}$
.
We sometimes call thebosonsystem therelevant system and the reservoir
system the irrelevant system.
The state of the system is described by the state vector $|\psi_{f}(t)\rangle\rangle$ in the space $\mathcal{H}_{S}^{0}\otimes$ $\Gamma^{\beta}$
.
The state vector$|\psi_{f}(t)\rangle\rangle$ is assumed to evolve in time according to the Schr\"odinger
equation
$d|\psi_{f}(t)\rangle\rangle=-i\mathcal{H}_{f,t}dt|\psi_{f}(t)\rangle\rangle$, (100)
with an infinitesimal time-evolution generator $\mathcal{H}_{f,t}$ includingrandom force operators. We
call the equation (100) stochastic Schr\"odinger equation.
The formal solution of (100) is written by
$|\psi_{f}(t)\rangle\rangle=V_{J}-(t)|\psi f(0)\rangle\rangle$, (101)
where $V_{f}(t)$ is the stochastic time-evolution generator satisfying the equation
with $V_{f}(0)=1$
.
Note that the $\mathrm{b}\mathrm{r}\mathrm{a}$-vector $\langle\langle$$\psi_{f}(t)|$ is defined by$\langle\langle$$\psi_{f}(t)|=\langle\langle\psi_{f}(0)|V_{f}^{-1}(t)$, (103)
where $V_{f}^{-1}(t)$ is the inverse of $V_{f}(t)$.
For $\mathrm{b}\mathrm{i}$-linear and phase invariant
boson system with the interaction
$i\sqrt{2\kappa}(a^{\dagger_{dB_{t}-a}\uparrow}dBt)$,
$\mathcal{H}_{f^{t}},dt$ has the form
$\mathcal{H}_{f,t}dt=Zdt+i\sqrt{2\kappa}(a^{\dagger_{dB_{t}d}}-aB^{\dagger}t)$ , (104)
with $Z\in \mathcal{H}_{S}^{0}\otimes\Gamma^{\beta}$ being operators having the forms
$Z=z_{s}\otimes I$, (105)
where $Z_{S}$ are a operator on $\mathcal{H}_{S}^{0}$ and $I$ is the identity operator on $\Gamma^{\beta}$.
$dB_{t},$ $dB_{t}^{1}$ are the
increment of the quantum Wiener processes at finite temperatures and $\kappa$ is a positive
$\mathrm{c}$-number. Note that we adopt the same notations for $B_{t},$ $B_{\mathrm{t}}^{\mathrm{f}}$ and
their tilde conjugates $\tilde{B}_{t},\tilde{B}_{t}^{\uparrow}$ as (63) and (64). In thefollowing, wewill put
$\overline{n}$to the Planck distribution function
(57).
Note that since the equation (102) with the infinitesimal time-evolution generator
(104) is the quantum stochastic differential equation ofthe Ito type, the time-evolution
generator $V_{f}(t)$ is the quantum stochastic integral of the Ito type which is an adapted
process.
We require that the time-evolution generator $V_{f}(t)$ should be unitary, i.e.
$V_{f}^{\dagger\uparrow(t)}(t)V_{f}(t)=V_{f}(t)Vf=1$. (106)
Therefore, we have the algebraic identities
$d[V_{f}^{\uparrow_{(t)V(t)}}f]=dV_{f}^{\uparrow}(t)\cdot V_{f}(t)+V_{f}^{\uparrow}(t)\cdot dV_{f}(t)+dV_{f}^{\uparrow}(t)dVf(t)=0$, (107)
and
$d[V_{j}(t)V^{\dagger}(ft)]=dV_{f}(t)\cdot V_{j}\uparrow(t)+V_{f}(t)\cdot dV_{f}^{\uparrow}(t)+dV_{f}(t)dV_{f}^{\uparrow}(t)=0$, (108)
where we have madeuse of the calculus rule of the Ito type (quantumIto’s formula). The
identities (107) and (108) with the equation (102) and its hermitian conjugates give the
following relation
where use has been made ofthe product rules (52). Thus, we obtain
74$f,tdt\overline{\infty}HSdt-i\kappa[(\overline{n}+1)a^{\dagger_{a}}+\overline{n}aa^{\uparrow}]dt+i\sqrt{2\kappa}(ad\uparrow B_{tt)}-adB^{\mathrm{t}}$
,
(110)where
we
put $(Z+z\uparrow)/2=H_{S}$.
Note that $H_{S}$ is hermitian.Applying the state vector $|\psi_{f}(0)\rangle\rangle$ to the equation (102) of$V_{f}(t)$, wehave the stochastic
Schr\"odinger equation ofthe Ito type
$d|\psi_{f(t)\rangle}\rangle=-i\mathcal{H}_{f,t}dt|\psi f(t)\rangle\rangle$, (111)
with the infinitesimal time-evolution generator (110).
5.2
The Stratonovich Type
Using the relation (85) between the Ito and the Stratonovich products, we transform the
stochastic differential equation (102) of the Ito type into that of the Stratonovich type as
$dV_{f}(t)=-i\mathcal{H}f,tdtVf(t)$
$=-i \{\mathcal{H}_{f,tf}dt\circ V(t)-\frac{1}{2}\mathcal{H}_{f,t}dtdVf(t)\}$
$\equiv-iH_{f^{\mathrm{g}}},dt\circ V_{f}(t)$, (112)
wherewe have substituted (102) into the right hand side of thesecond equality. Here, we
defined the infinitesimal time-evolution generator $H_{f,t}$ of the Stratonovich type by
$H_{f,t}dt \equiv \mathcal{H}_{f,t}dt+i\frac{1}{2}\mathcal{H}_{f,t}dt\mathcal{H}f,tdt$
.
(113)With the help ofthe product rules (52),
we
obtain the hermitian stochastic infinitesimaltime.evolution generator $H_{f,t}dt$as
$H_{f,t}dt=H_{S}dt+i\sqrt{2\kappa}(a^{\uparrow}dB_{t}-adB_{t}^{\uparrow})$ . (114)
The hermiticy of $H_{f,t}dt$ guarantees the unitarity of $V_{f}(t)$
.
Applying the state vector $|\psi_{f}(\mathrm{o})\rangle\rangle$ to theequation (112) of$V_{f}(t)$, weobtain the
stochas-tic Schr\"odinger equation of theStratonovich type
$d|\psi_{f}(t)\rangle\rangle=-iH_{f,t}dt\circ|\psi_{f}(t)\rangle\rangle$, (115)
6Stochastic
Time-Evolution in Thermal
Space
On
the basis of the stochastic Schr\"odinger equation, investigated in the previous section,we will construct astochastic Liouville equation in thermal space and obtain the explicit
form of the time-evolution generator satisfying the stochastic Liouville equation within
the frameworkof NETFD. Using the time-evolutiongenerator, wewill construct a unified
canonical operator formalism of quantum stochastic differential equations.
6.1
Thermal Vacuums
Let us define the density operator $\rho_{f}(t)$ corresponding to the state vector $|\psi_{f}(t)\rangle\rangle$ by
$\rho_{f}(t)\equiv|\psi_{f}(t)\rangle\rangle\langle\langle\psi_{f}(t)|=V_{f}(t)|\psi_{f}(0)\rangle\rangle\langle\langle\psi f(\mathrm{o})|V_{f}^{\dagger}(t)=V_{f}(t)\rho f(0)Vf\dagger(t)$, (116)
where use has been made of (101) and (103) with the unitary time-evolution generator
$V_{f}(t)$. The density operator $\rho_{f}(t)$ satisfies
$\mathrm{t}\mathrm{r}_{tot}\rho_{f}(t)=1$, (117)
where the trace operation $\mathrm{t}\mathrm{r}_{tot}$ is defined by
$\mathrm{t}\mathrm{r}_{tt}\circ\equiv \mathrm{t}\mathrm{r}\otimes \mathrm{t}\mathrm{r}_{R}$, (118)
with the trace operations tr ofthe relevant system and $\mathrm{t}\mathrm{r}_{R}$ ofthe reservoir. The
expecta-tion value ofany observable $A$ is given by $\mathrm{t}\mathrm{r}_{to}tA\rho f(t)$.
With thehelpofthe principleofcorrespondence (seeappendixA),the density operator
$\rho_{f}(t)$ defined by (116) is expressed as a thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum, i.e.
$|0_{f}(t)\rangle\equiv|\rho_{f}(t)\rangle=\hat{V}_{f}(t)|0_{f}(\mathrm{o})\rangle$, (119)
where we have defined the stochastic timeevolutiongenerator by
$\hat{V}_{f}(t)=V_{f}(t)\tilde{V}_{f}(t)$. (120)
Note that, since $V_{f}(0)=1$, we have $\dot{V}_{f}(0)=1$
.
The vector space to which the thermalvacuum $|0_{f}(t)\rangle$ belongs is assumed to be$\mathcal{H}_{S}\otimes\Gamma^{\beta}$ where$\mathcal{H}_{S}$is thespace ofrelevantsystem
and $\Gamma^{\beta}$ is
the Fock space of the system of random force operators constructed in section
3. The operator $\hat{V}_{f}(t)$ defined by (120) is on the space $\mathcal{H}_{S}\otimes\Gamma^{\beta}$ and tums out to be
unitary from the relation
where use has been made ofthe unitarity of $V_{f}(t)$
.
Note that the space ofstates $\mathcal{H}_{S}$ ofrelevant system is expressed as $\mathcal{H}_{S}=\mathcal{H}_{s}^{0}\otimes\tilde{\mathcal{H}}_{S}^{0}$ with the usual Hilbert space $\mathcal{H}_{S}^{0}$ for wave function and its tilde conjugate space $\tilde{\mathcal{H}}_{S}^{0}$.
The equation (117) requires that
$\langle 1_{tot}|0_{f}(t)\rangle=1$, (122)
where the thermal $\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum
$\langle$$1_{tot}|$ is defined by
$\langle$$1_{tot}|\equiv\langle|\langle 1|$, (123)
with the thermal $\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum
$\langle$$1|$ in the space $\mathcal{H}_{S}$ of the relevant system and the thermal
$\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum $\langle$$|$ in the space
$\Gamma^{\beta}$
of the irrelevant system. Theexpectationvalue$\mathrm{t}_{\Gamma_{tot}}A\rho f(t)$
is expressed as the expectation with respect to the thermal $\mathrm{k}\mathrm{e}\mathrm{t}$
-vacuum
$|0_{f}(t)\rangle$ and the
thermal $\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum
$\langle$$1_{tot}|$, i.e.
$\langle 1_{tot}|A|0f(t)\rangle=\mathrm{t}\mathrm{r}_{tot}A\rho f(t)$. (124)
Note that for any relevant system operator $A$, we have
$\langle$$1|A\dagger=\langle 1|\tilde{A}$, (125)
which is the basic property of thermalspace $[17]-[19]$. Furthermore, for the random force
operators $dB_{t},$ $dB_{t}^{\uparrow}$, we have
$\langle$$|dB_{t}\dagger=\langle|d\tilde{B}_{t}$, (126)
which follows from (29).
The equation (122) together with (119) yields
$\langle 1_{tot}|\hat{V}_{J^{-()1}}t0f(0)\rangle=1$. (127)
Since
the $\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}}(127)$ should hold for any time $t$ and forany
initial thermalvacuum
$|0_{f}(\mathrm{o})\rangle$, we have
$\langle$$1_{tot}|\hat{V}_{f(t)}=\langle 1_{tot}|\hat{V}f(0)=\langle 1_{tot}|$, (128)
6.2
Stochastic Liouville
Equation
6.2.1 The $\mathrm{I}\mathrm{t}\mathrm{o}\dot{\mathrm{T}}\mathrm{y}\mathrm{p}\mathrm{e}$Using the calculus rule of the Ito type,
we
have from (120)$d\hat{V}_{f}(t)=dV_{f}(t)\cdot\tilde{V}_{f(t})+V_{f}(t)\cdot d\tilde{V}_{f}(t)+dV_{f}(t)d\tilde{V}f(t)$
.
(129)Substituting (102) and its tilde conjugate:
$d\tilde{V}_{f}(t)=i\tilde{\mathcal{H}}f,tdt\tilde{V}_{f}(t)$, (130)
into (129), we have
$d\hat{V}_{f}(t)=-i\hat{\mathcal{H}}f,tdt\hat{V}f(t)$, (131)
where
$\hat{\mathcal{H}}_{f,t}dt\equiv \mathcal{H}_{f,t}dt-\tilde{\mathcal{H}}_{f,t}dt+i\mathcal{H}_{f,t}dt\tilde{\mathcal{H}}_{f},tdt$
.
(132)With the help of (110) and the product rules (52), $\mathcal{H}_{f,t}dt\tilde{\mathcal{H}}f,tdt$ is calculated as
$\mathcal{H}f,tdt\tilde{\mathcal{H}}f,tdt=2\kappa[(\overline{n}+1)a\tilde{a}+\overline{n}a^{\mathrm{t}}\tilde{a}\dagger]dt$. (133)
Putting (110) and (133) into (132), we obtain
$\hat{\mathcal{H}}_{f,t}dt=\hat{H}_{S}dt+i(\wedge\Pi_{R}+\Pi_{D)dt}\wedge+d\hat{M}_{t},$ (134) where $\hat{H}_{s=}H_{S}-\tilde{H}s$, (135) $\hat{\Pi}_{R}=-\kappa[(a^{\uparrow}-\tilde{O})(\mu a+\nu\tilde{a}^{\mathrm{t}})+\mathrm{t}.\mathrm{c}.]$ , (136) $\hat{\Pi}_{D}=2\kappa(\overline{n}+\nu)(a^{\uparrow_{-\tilde{a}}})(\tilde{a}\uparrow-a)$, (137) and $d\hat{M}_{t}=i\{[(a^{\mathrm{t}}-\tilde{a})dWt+\mathrm{t}_{\mathrm{C}}..]-[(\mu a+\nu\tilde{a}^{\mathrm{t}})dW_{\mathrm{t}}^{\}+\mathrm{t}.\mathrm{c}.]\}$, (138)
with real numbers $\mu,$ $\nu$ satisfying $\mu+\nu=1$. The operators $dW_{t}$ and $dW_{t}^{*}$
are
defined by$dW_{t}\equiv\sqrt{2\kappa}(\mu dB_{t}+\nu d\tilde{B}_{t}^{\dagger})$
,
$dW_{t}^{\}\equiv\sqrt{2\kappa}(dB_{tl)}^{\uparrow_{-}d\tilde{B}}$. (139)Making use of the relations (125) and (126), we seethat (134) satisfies
which is consistent with the relation (128). Note that
$\langle$$1|\hat{\mathcal{H}}_{f,t}dt\neq 0$, (141)
which indicates that the conservation of probability does not hold within only the space of states of relevant system, i.e.
$\langle 1|0_{f}(t)\rangle\neq 1$
.
(142)Similarly, from the definition
$\hat{V}_{f}^{\uparrow}(t)=V^{\mathrm{t}_{(t}}f)\tilde{V}^{\dagger}f(t)$, (143) we obtain $d\hat{V}_{f}^{\mathrm{t}}(t)=i\hat{V}_{f}^{\uparrow t)t}(\hat{\mathcal{H}}_{f}(,-1)d\mathrm{t}$ ’ (144) $\backslash \mathrm{v}\mathrm{i}\mathrm{t}\mathrm{h}$ $\hat{\mathcal{H}}_{f,i}^{(-1})dt=\hat{H}_{S}dt-i(\wedge\Pi_{R}+\Pi_{D})\wedge dt+d\hat{M}_{\mathrm{t}}$. (145)
We see that the equations (131) with (134) and (144) with (145) satisfy
$d\hat{V}_{f}^{\uparrow}(t)\cdot\hat{V}_{f}(t)+\hat{V}_{f}^{1}(t)\cdot d\hat{V}_{f}(t)+d\hat{V}_{f}^{\mathrm{t}}(t)d\hat{V}_{f}(t)=0$, (146)
and
$d\hat{V}_{f}(t)\cdot\hat{V}_{f}^{\dagger}(t)+\hat{V}_{f}(t)\cdot d\hat{V}_{f}^{\uparrow}(t)+d\hat{V}_{f(t)d\hat{V}_{f}^{\uparrow}(t})=0$, (147)
which are consistent of the unitarity of $\hat{V}_{f}(t)$.
$.l$
Since $\hat{V}_{f}(t)$ and $\hat{V}_{f}^{\uparrow}(t)$ are subject to the stochastic differential equations (131) with
(134) and (144) with (145) of the Ito type, respectively, they are quantum stochastic
processes consisting of quantumstochastic integrals ofthe Ito type. Therefore, $\hat{V}_{f}(t)$ and
$\hat{V}_{j}^{1}(t)$ are adapted processes.
Applying the state vector the thermal vacuum $|0_{f}(0)\rangle$ to the equation (131) of$\hat{V}_{f}(t)$,
we obtain the quantum stochastic Liouville equation of the Ito type
$d|0_{f}(t)\rangle=-i\hat{\mathcal{H}}_{f^{t}},dt|\mathrm{o}_{f(}t)\rangle$ , (148)
$\backslash \mathrm{v}\mathrm{i}\mathrm{t}\mathrm{h}$the infinitesimal
6.2.2 The Stratonovich Type
Using the calculus rule of the Stratonovich type,
we
have from (120)$d\hat{V}_{f}(t)=dV_{f}(t)\circ\tilde{V}f(t)+V_{f}(t)\circ d\tilde{V}_{f}(t)$
.
(149)Substituting (112) and its tilde conjugate
$d\tilde{V}_{f}(t)=i\tilde{H}\mathrm{t}dt\circ f,\tilde{V}f(t)$, (150)
into (149), we obtain
$d\hat{V}_{f}(t)=-i\hat{H}_{f,f}tdt\circ\hat{V}(t)$, (151)
where
$\hat{H}_{f,t}dt\equiv H_{f,t}dt-\tilde{H}_{f,\mathrm{t}}dt$
.
(152)Putting (114) into (152), we get
$\hat{H}_{f,t}dt=\hat{H}_{S}dt+d\hat{M}_{t}$, (153)
which is apparently hermitian.
Taking the hermitian conjugate ofthe equation (151), we have the equation of $\hat{V}_{f}^{1}(t)$
as
$d\hat{V}_{f}^{1}(t)=i\hat{V}_{f}^{1}(t)\circ\hat{H}f,tdt$, (154)
where use has been made of the hermiticy of$\hat{H}_{f,t}dt$.
The equations (151) and (154) with (153) satisfy the following equations
$d\hat{V}_{f}^{\dagger}(t)\circ\hat{V}f(t)+\hat{V}_{f}^{\uparrow_{(t)\circ}t)}d\hat{V}f(=0,$
$(155)$
$d\hat{V}_{f}(t)\circ\hat{V}_{f}\mathrm{t}_{(t})+\hat{V}_{f}(t)\circ d\hat{V}_{f}^{\uparrow()}t=0$, (156)
which show the unitarity of $\hat{V}_{f}(t)$.
With the help of the properties (125) and (126), we see that the expression (153)
satisfies
$\langle$$1_{tot}|\hat{H}f,\iota^{dt}=\langle|\langle 1|\hat{H}_{f,t}dt=0$, (157)
which is consistent with (128).
The time-evolution equation (131) of the Ito type with (134) is connected to the
equation (151) of the Stratonovich type with (153) by the relation (85) between the Ito
(145) is connected to the equation (154) ofthe Stratonovich type with (153) through the
relation (84).
Applying the state vector the thermal
vacuum
$|0_{f}(0)\rangle$ to the equation (151) of $\hat{V}_{f}(t)$,we
have the quantum stochastic Liouville equation of the Stratonovich type$d|0_{f}(t)\rangle=-i\hat{H}_{f,t}dt\circ|0_{f}(t)\rangle$, (158) $\backslash \mathrm{v}\mathrm{i}\mathrm{t}\mathrm{h}$ the infinitesimal time-evolution generator
(153).
6.3
Fokker-Planck
Equation
Applying the random force $\mathrm{b}\mathrm{r}\mathrm{a}$
-vacuum
$\langle$$|$ to the stochastic Liouville equation (148) of
the Ito type with the infinitesimal time-evolution generator (134), we have
$d\langle|0_{f}(t)\rangle=-i\langle|\hat{\mathcal{H}}_{f,t}dt|\mathrm{o}f(t)\rangle$
$=-i\{\hat{H}dt\langle|\mathrm{o}f(t)\rangle+\langle|d\hat{M}_{t}|\mathrm{o}_{f}(t)\rangle\}$
.
(159)Under the assumption
$|0_{f}(\mathrm{o})\rangle=|\mathrm{o}_{S}\rangle|\rangle$, (160)
with the thermal
vacuum
$|0_{S}\rangle$ of relevant system at $t=0,$ $\langle|d_{\mathit{1}}|\wedge/_{I_{t}|0}f(t)\rangle$ can be evaluatedas
$\langle|d\hat{M}_{t}|\mathrm{o}_{f(t})\rangle=0$, (161)
where we used the definition (119) of the thermal vacuum $|0_{f}(t)\rangle$ and the property (79)
of the products of the Ito type. Therefore, putting $|0(t)\rangle=\langle|0_{f}(t)\rangle$, we finally obtain the
Fokker-Planck equation for $\mathrm{b}\mathrm{i}$-linear
and phase invariant system as
$\frac{\partial}{\partial t}|0(t)\rangle=-i\hat{H}|\mathrm{o}(t)\rangle$ ,
(162)
where the infinitesimal time-evolution generator $\hat{H}$
is given by
$\hat{H}=\hat{H}_{S}+i\hat{\Pi}$, (163)
with
$\hat{\Pi}=\hat{\Pi}_{R}+\hat{\Pi}_{D}$
which is identical to that obtained within the framework of NETFD $[17]-[19]$
.
Note thatwe can also derive the Fokker-Planck equation (162) by applying the random force
bra-vacuum
$\langle$$|$ to the stochastic Liouville equation (158) of the Stratonovich type with theinfinitesimal time-evolution generator (153) $[17, 20]$.
Recalling the equation (119) and taking (160) into account, we find that
$|0(t)\rangle=\langle|\hat{V}_{f}(t)|\rangle|0_{S}\rangle$
.
(165)On the other hand, the timeevolution generator $\hat{V}(t)$ ofthe thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum $|0(t)\rangle$ is
defined by
$|0(t)\rangle=\hat{V}(t)|0(0)\rangle$
.
(166)Provided that $|0(0)\rangle=|0_{S}\rangle$, the equations (165) and (166) yield
$\hat{V}(t)=\langle|\hat{V}_{f}(t)|\rangle$. (167)
6.4
Quantum Langevin Equation
Since $\hat{V}_{f}(t)$ is unitary, we have
$\hat{V}_{f}^{1}(t)\hat{V}_{f}(t)=\hat{V}_{f}(t)\hat{V}_{f}\uparrow(t)=1$. (168)
From the equation (168), we see with the help of the property (128) that
$\langle$$1_{tot}|\hat{V}^{\uparrow()=}ft\langle 1_{tot}|$. (169)
The expectation valueofanyobservable $A$with respect to the state $|0_{f}(t)\rangle$ is given by
$\langle 1_{to}\iota|A|0f(t)\rangle=\langle 1_{tot}|A\hat{V}_{f(}t)|\mathrm{o}_{f}(0)\rangle$
$=\langle 1_{tot}|\hat{V}f\uparrow_{(t})A\hat{V}_{f}(t)|0f(0)\rangle$, (170)
wherewe have used the equation (119) and the property (169). Ifwe define the operator
in the Heisenberg representation
$A(t)=\hat{V}_{f}^{1}(t)A\hat{V}_{f(t})_{\mathrm{i}}$ (171)
we consider (170) to be the expectation value of $A(t)$ with respect to the initial state $|0_{f}(\mathrm{o})\rangle$. Note that, as $\hat{V}_{f}(t)$ and $\hat{V}_{f}^{\dagger}(t)$ are adapted processes, the operator $A(t)$ defined by (171) is also an adapted process. Therefore, the following commutation relation holds:
for quantumWiener processes$B_{t},$ $B_{t}^{\mathrm{t}}$ and their tildeconjugates $\tilde{B}_{t},\tilde{B}_{t}^{\mathrm{t}}$, whichcomesfrom
(76).
Any operators in the Heisenberg representation defined by (171) keep the equal-time
commutation relations, such as
$[a(t), a^{\uparrow}(t)]$
.
$=1,$. $[\tilde{a.}(.t),\tilde{a}^{\uparrow}(t.)]=\backslash 1$
.
(173)Note that, using the properties (125) and (169), we have for$A(t)$ defined by (171)
$\langle$$1_{tot}|A\dagger(t)=\langle 1_{tot}|\tilde{A}(t)$
.
(174)Using thecalculus rule of the Ito type, we have thealgebraic identity for the operator
$A(t)$ defined by (171)
$dA(t)=d\hat{V}_{f}^{\dagger}(t)A\hat{V}_{f}(t)+\hat{V}_{f}^{1}(t)Ad\hat{V}f(t)+d\hat{V}_{f}^{\uparrow t)}(t)Ad\hat{V}f$( . (175)
Substituting the equations (131) with (134) and (144) with (145) into (175), we obtain
the quantum Langevin equation of the Ito type.
On theother hand, making useofthe calculus rule ofthe Stratonovich type, we have
the algebraic identity for the operator $A(t)$ defined by (171)
$dA(t)=d\hat{V}_{f}\uparrow(t)\circ A\hat{V}f(t)+\hat{V}_{f}^{\uparrow_{(t}t)})A\circ d\hat{V}f$( . (176)
With the help of the equation (151) and its hermitian conjugate (154) together with the
identity (176), we obtain the quantum Langevin equation of the Stratonovich type. We
seethat the quantum Langevin equationof the Stratonovich type canbeexpressed as the
Heisenberg equation for$A(t)$:
$dA(t)=i[\hat{H}_{f}(t)dt^{\circ},$ $A(t)]$ , (177)
where we defined
$\hat{H}_{f}(t)dt=\hat{V}_{f}^{\uparrow}(t)\circ\hat{H}_{f},tdt\circ\hat{V}f(t)$
.
(178)The symbol $[\cdot\circ, \cdot]$ is the commutator defined by (80).
6.5
Equation of
Motion
of
Expectation Value
Let us
assume
that the initial vacuum $|0_{f}(\mathrm{o})\rangle$ can be expressed by the product of thevacuums
ofthe relevant and the irrelevant systems aswhere $|0_{S}\rangle$ $\in \mathcal{H}_{S}$ is the thermal
$\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum of the relevant system at time $t=0$
.
Applying the $\mathrm{b}\mathrm{r}\mathrm{a}$-vacuum $\langle$$1_{\mathrm{t}ot}|$ to the quantum Langevin equation ofthe Ito type, we
have
$d\langle 1_{t_{\mathit{0}}t}|A(t)=i\langle 1_{tot}|[H_{S}(t), A(t)]dt$
$+\kappa(\langle 1_{tot}|a^{\dagger}(t)[A(t), a(t)]+\langle 1_{tot}|[a^{\uparrow}(t),$ $A(t)]a(t))dt$
$+2\kappa\overline{n}\langle 1tot|[a^{\dagger}(t),$ $[A(t), a(t)]]dt$
$+\langle 1_{tot}|[A(t),$ $a^{\uparrow}(t)]dFt+\langle 1_{tot}|[a(t), A(t)]dF_{t}\uparrow$, (180)
where we used the properties (126) and (174).
Putting the $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum
$|0_{f}(\mathrm{o})\rangle\equiv|0_{f}\rangle$ to (180), we obtain the equation of motion of
the expectation value of an arbitrary operator $A$ of the relevant system as
$\frac{d}{dt}\langle 1_{tot}|A(t)|0_{f}\rangle=i\langle 1_{tot}|[H_{S}(t), A(t)]|0_{f}\rangle$
$+\kappa(\langle 1_{t_{\mathit{0}}t}|a^{\dagger}(t)[A(t), a(t)]|0_{f}\rangle+\langle 1_{tot}|[a^{\mathrm{t}}(t), A(t)]a(t)|0_{f}\rangle)$
$+2\kappa\overline{n}\langle 1t_{ot}|[a^{\dagger}(t), [A(t), a(t)]]|0_{f}\rangle$. (181)
Here, we used the property (79) of the Ito products.
Remembering (167) and the definition (171) of $A(t)$, we find with the assumption
(160) that
$\langle 1_{tot}|A(t)|0_{f}\rangle=\langle|(1|\hat{V}f\uparrow(t)A\hat{V}f(t)|0s\rangle|\rangle=\langle 1|A|0(t)\rangle$, (182)
where we have used the property (169) and the assumption that $|0(0)\rangle=|0_{S}\rangle$. Taking
account of the relation (182), we see that the equation (181) of expectation value is
identicalto the equation derived fromtheFokker-Planck equation (162) with (163), (164),
which shows the consistency ofthe framework.
7
Summary
and
Discussion
In this paper, weconstructed the quantum Wiener processes togetherwith their
represen-tation space by extending the work of mathematicians and by implanting it into NETFD.
Then, weconstructed a unified system of quantum stochastic differential equations on the
The quantum Wiener processes, which we employed as random force operators, are constructed by using boson operators with time indices together with their
representa-tion space. When we adopted the Fock space $\Gamma^{0}$ for the representation space, in the
same way as Hudson and Parthasarathy, we obtained the quantum Wiener processes at
zero temperature (section 2). Whereas, we obtained the quantum Wiener processes at
finite temperatures by extending the representation space to the Fockspace $\Gamma^{\beta}$
which is
obtained by the Bogoliubov transformation in the tensor product space $\Gamma=\Gamma^{0}\otimes\tilde{\Gamma}^{0}$
.
There, the thermal degree of freedom was introduced by the thermal state conditions
or the Bogoliubov transformation, which is a manifestation of the unitary inequivalence
between the thermal vacuums ofzero and finite temperatures (section 3). This notion
ofthe unitary inequivalence between the vacuums with different temperatures is one of
the remarkable features within NETFD or TFD. The quantum Wienerprocesses and the
quantum stochastic calculus given in this paper provide the foundation for those used in
quantum optics [3] and quantum stochastic differential equations within NETFD $1^{20}1-[30]$
.
We
constructed the stochastic Schr\"odinger equation with the quantum Wienerpro-cesses at finite temperatures on the requirement that the time-evolution generator should
be unitary (section 5). Then, we introduced the density operator corresponding to the
stochasticwavefunction. Bymeans of the principle ofcorrespondence between quantities
in the thermal space and in the Hilbert space, we obtained the stochastic thermal
ket-vacuum corresponding to the density operator and a stochastic Liouville equation. On
the basis of the timeevolution of the thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum, we constructed the
system of
quantum stochastic differential equations wvithin NETFD (section 6).
The timeevolution equation of the thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum gave
the quantum stochastic
Liouville equation. On the other hand, the Heisenberg equation with the infinitesimal
time-evolution generator of the quantum stochastic Liouville equation gave the
quan-tum Langevin equation. Using the quantum stochastic calculus constructed in section 4,
we constructed the quantum stochastic differential equations both ofthe Ito and of the
Stratonovich types.
Applying the random force $\mathrm{b}\mathrm{r}\mathrm{a}$
-vacuum
$\langle$$|$ to the stochastic Liouville equation of theIto type,
we
obtained the Fokker-Planck equation, which is identical to that derived inthe papers $[17]-[19]$
Taking the expectation with respect to the thermal $\mathrm{k}\mathrm{e}\mathrm{t}$-vacuum $|0_{f}\rangle$ and the thermal
equation of motion of expectation value ofan arbitrary relevant system operator. This equation of motion is equivalent to that derived by the Fokker-Planck equation, which
shows the self-consistency ofthe system.
Hudson and Lindsay constructed a unitary stochastic time-evolution in the vector
space $\mathcal{H}_{S}^{0}\otimes\Gamma^{\beta}$, where $\mathcal{H}_{S}^{0}$ for relevant system is ausual Hilbert space and $\Gamma^{\beta}$ for random
force operators is a Fock space in thermal space, which was briefly reviewed in section
5. The fact that the vector space for relevant system is not a thermal space prevents the
system from introducing the quantum stochastic Liouville equation. In this paper, we
completed the motivation of Hudson and Lindsay by adopting a thermal space for the
space of states ofrelevant system
as
wellas
random force system and giving the quantumstochastic Liouville equation.
The stochastic time-evolutionconstructedinthis paper is unitary. On theother hand,
non-unitary stochastic time-evolution was constructed within the framework of NETFD
$[20]-[30]$. In this way, it turned out that there exist two kinds of systems in quantum
stochasticdifferential equations within NETFD, one of them is the systemof non-unitary
stochastic time-evolution and the other $1\mathrm{S}$ that of unitary stochastic time-evolution. The
two systems
are
equivalent in the sense that they give the same equation of motionof expectation value of any observable. The relation between the two systems will be
investigated in the forthcoming paper.
A
The
Principle
of
Correspondence
The correspondence between vectors in thethermal space andoperators in a Hilbert space
is given by the followving rule [37, 38, 39]:
$\rho_{S}(t)$ $rightarrow$ $|\mathrm{o}(t)\rangle$, (183)
$A_{1}\rho_{S}(t)A_{2}$ $rightarrow$ $A_{1}\tilde{A}_{2}|0\dagger(t)\rangle$. (184)
Here, $\rho_{S}(t)$ is a density operator on the Hilbert space, whereas $|0(t)\rangle$ is a thermal
ket-vacuum in the thermal space. $A_{1},$ $A_{2}$ are arbitrary operators on the Hilbert space.
It was noticed first by Crawford [40] that the introduction oftwo kinds of operators
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